Mathematics

Trigonometric Identities and Equations

223 Questions

Solve a variety of questions based on trigonometric identities and mathematical equations. Key areas covered include double angle formulas, the law of sines, and quadrant angles. This material helps students preparing for advanced mathematics exams, UPSC, and state public service commissions.

Double angle formulasLaw of sinesTrigonometric quadrantsLaplace transformSecant functionTaylor series

Trigonometric Identities and Equations Questions

Multiple choice

Find all solutions of the equation (\tan^2\theta + \sec\theta - 1 = 0) in the interval ([0, 2\pi)).

  1. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the identity (\sec^2\theta = 1 + \tan^2\theta), we can rewrite the equation as (1 + \tan^2\theta + \tan\theta - 1 = 0). Simplifying, we get (\tan^2\theta + \tan\theta = 0). Factoring, we find (\tan\theta(\tan\theta + 1) = 0). Solving each factor separately, we find (\tan\theta = 0) or (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Multiple choice

Solve the equation (\sin^2\theta + \cos^2\theta - \sin\theta - \cos\theta = 0) for (0 \le \theta \le 2\pi).

  1. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the identity (\sin^2\theta + \cos^2\theta = 1), we can simplify the equation to (1 - \sin\theta - \cos\theta = 0). Rearranging, we get (\sin\theta + \cos\theta = 1). This suggests that (\sin\theta) and (\cos\theta) have the same sign. The only way this can happen is if both (\sin\theta) and (\cos\theta) are positive. Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Multiple choice

Solve the equation (2\sin^2\theta - 3\sin\theta + 1 = 0) for (0 \le \theta \le 2\pi).

  1. \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\)
  2. \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)
  3. \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)
  4. \(\theta = 0, \pi\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Multiple choice

What is the value of (\sin 90^\circ) in Indian trigonometry?

  1. 0

  2. 1
  3. \(\frac{1}{2}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In Indian trigonometry, the value of (\sin 90^\circ) is 1.

Multiple choice

What is the value of (\cos 0^\circ) in Indian trigonometry?

  1. 0

  2. 1
  3. \(\frac{1}{2}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In Indian trigonometry, the value of (\cos 0^\circ) is 1.

Multiple choice

The reciprocal of the tangent of an angle is called the:

  1. Cosecant

  2. Secant

  3. Cotangent

  4. Cosine
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cotangent of an angle is defined as the reciprocal of the tangent of the angle.

Multiple choice

The trigonometric ratio that relates the length of the opposite side to the length of the hypotenuse is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Secant
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Multiple choice

The trigonometric ratio that relates the length of the adjacent side to the length of the hypotenuse is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Secant
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Multiple choice

The trigonometric ratio that relates the length of the hypotenuse to the length of the adjacent side is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Secant
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The secant of an angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side.

Multiple choice

What is the sine rule?

  1. \(\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}\)
  2. \(\frac{sin A}{a} = \frac{cos B}{b} = \frac{tan C}{c}\)
  3. \(\frac{cos A}{a} = \frac{sin B}{b} = \frac{tan C}{c}\)
  4. \(\frac{cos A}{a} = \frac{cos B}{b} = \frac{sin C}{c}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine rule states that in a triangle, the ratio of the sine of an angle to the length of the opposite side is the same for all angles.

Multiple choice

What is the cosine rule?

  1. \(c^2 = a^2 + b^2 - 2ab cos C\)
  2. \(c^2 = a^2 + b^2 + 2ab cos C\)
  3. \(c^2 = a^2 - b^2 + 2ab cos C\)
  4. \(c^2 = a^2 - b^2 - 2ab cos C\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cosine rule states that in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle between them.

Multiple choice

What is the double angle formula for sine?

  1. \(sin 2A = 2 sin A cos A\)
  2. \(sin 2A = sin A + sin A\)
  3. \(sin 2A = cos A + cos A\)
  4. \(sin 2A = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double angle formula for sine states that (sin 2A = 2 sin A cos A).

Multiple choice

What is the double angle formula for cosine?

  1. \(cos 2A = cos^2 A - sin^2 A\)
  2. \(cos 2A = cos A + cos A\)
  3. \(cos 2A = sin A + sin A\)
  4. \(cos 2A = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double angle formula for cosine states that (cos 2A = cos^2 A - sin^2 A).

Multiple choice

What is the half angle formula for sine?

  1. \(sin \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{2}}\)
  2. \(sin \frac{A}{2} = sin A + sin A\)
  3. \(sin \frac{A}{2} = cos A + cos A\)
  4. \(sin \frac{A}{2} = 2 sin A cos A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half angle formula for sine states that (sin \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{2}}).

Multiple choice

What is the half angle formula for cosine?

  1. \(cos \frac{A}{2} = \pm \sqrt{\frac{1 + cos A}{2}}\)
  2. \(cos \frac{A}{2} = cos A + cos A\)
  3. \(cos \frac{A}{2} = sin A + sin A\)
  4. \(cos \frac{A}{2} = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half angle formula for cosine states that (cos \frac{A}{2} = \pm \sqrt{\frac{1 + cos A}{2}}).